Geometry - Stereographic projection.

I know if a cirlce (on S^2) does not contain N (0,0,1) then it is mapped onto the plane H as a circle. Now say the circles on S^2 are lines of latitude. When mapped by the stereographic projection they are cirlces in R^3 on the plane H. Now the only thing im not sure on is,

my claim:
When lines of latitude are mapped by the stereographic projection the radius of the circle on H, is the same radies as the circle that was projected from S^2.

I know for one thing that if the equator is mapped onto H it is the same on H as on S^2.